( g,e ) -Symmetric Rings

Abstract

Let [Formula: see text] be a ring and [Formula: see text], [Formula: see text] in [Formula: see text], the set of idempotents of [Formula: see text]. Then [Formula: see text] is called [Formula: see text]-symmetric if [Formula: see text] implies [Formula: see text] for any [Formula: see text], [Formula: see text], [Formula: see text]. Clearly, [Formula: see text] is an [Formula: see text]-symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring; in particular, [Formula: see text] is a symmetric ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring. We show that [Formula: see text] is left semicentral if and only if [Formula: see text] is a [Formula: see text]-symmetric ring; in particular, [Formula: see text] is an Abel ring if and only if [Formula: see text] is a [Formula: see text]-symmetric ring for each [Formula: see text]. We also show that [Formula: see text] is [Formula: see text]-symmetric if and only if [Formula: see text], [Formula: see text] is symmetric, and [Formula: see text] for any [Formula: see text], [Formula: see text]. Using [Formula: see text]-symmetric rings, we construct some new classes of rings.